Operations: search;; min;; max;; predecessor;; successor. Time O(h) with h height of the tree (more on later).

Save this PDF as:
 WORD  PNG  TXT  JPG

Size: px
Start display at page:

Download "Operations: search;; min;; max;; predecessor;; successor. Time O(h) with h height of the tree (more on later)."

Transcription

1 Binary search tree Operations: search;; min;; max;; predecessor;; successor. Time O(h) with h height of the tree (more on later). Data strutcure fields usually include for a given node x, the following attributes: x.left (left child);; x.right (right child);; x.p (parent);; x.key. Root of entire tree pointed to by T.root Example of storing keys in a binary search tree: (binary because branches either to left or to right) Do you see a pattern? Main property of binary search tree: Left side: key smaller equal to parent Right side: key larger equal to parent Given node with key k Left: All keys < k Right: All keys > k You can see this in tree example above node by node going from parent to child, and it is also true for the whole subtree to the left and to the right. Is this tree unique for the given set of keys? Answer: No;; The same set of keys have many possible trees.

2 Another example: Question: How can we traverse the tree and print out keys in ascending order: First left side: Then root: Then right:

3 What about this one? (answer: same: ) Runtime for inorder-tree-walk? Phi(n) for a binary tree with n nodes Why? Omega(n) because visits every node of the tree at least once and print the key;; book shows also O(n) using subsitution method (it s intuitive in any case that you traverse all nodes no more than constant amount of times). Query operations on binary search tree: search;; max;; min;; successor;; predecessor. Run time depends on? Height of binary search tree. Question: Is the height always log n? Answer: Not necessarily, since tree could be made non efficiently and each time branch to the right only. Then its run time could be O(n) and as bad as a linked list. Therefore height and run time for search queries could be O(log n) but could also be O(n)! Search: Main idea of searching for key k: if looking for k smaller than node.key, search to the left, otherwise to the right. Searching for k = 13:

4 Search trajectory for k=13: 15 -> 6 -> 7 -> 13 Tree-Search(x,k) 1. if k=x.key or x==nil return x 2. if k < x.key return Tree-Search(x.left, k) 3. else if k > x.key return Tree-Search(x.right, k) (more formal in book and can also be written iterative in book) Run time: height of tree. Again, could be O(log n) or O(n) depending on tree and its balance. Min/max: Question: finding the min? Answer: keep traversing to the left subtree until we encounter a nil Finding the min: Finding the max: keep traversing to the right subtree

5 Successor: (similar for predecessor) Case 1: Right sub-tree non empty: left-most node of right subtree (why? It s the smallest value that is larger than the given node) Example case 1: successor of 15 Right sub-tree non empty: Left most node of right sub-tree Case 2: Right subtree is empty, then successor is an ancestor (we ll first define ancestor and then more specifically what we mean here) Ancestor definition in Cormen book: an ancestor of node x is any node from the root to node x. This is either a parent of x or the node x itself. More specifically: successor of node x is an ancestor that is the lowest ancestor of node x, whose left child is also an ancestor of x. Put differently: Example case 2: successor of 9 Right sub-tree empty: Successor of 9 is 13; lowest ancestor of 9 whose left child (here 9) is also an ancestor of 9

6 Another example case 2: successor of node 4 Right sub-tree empty: Successor of 4 is 6; lowest ancestor of 4 whose left child is also an ancestor of 4 Why does this work? Consider node x (here node 4). Intuitively, if left child of ancestor y is also an ancestor of node x (here node 6), then node x is smaller than the ancestor y. All ancestor nodes before y (here before node 6 = node 3) do not have a left child that is an ancestor of x, which means that x is a right child of that ancestor and so that ancestor is smaller than x. Another example case 2: successor of node 20 Answer: none. Why? Another example case 2: successor of node 13 is 15. Why? Run time of successor: O(height), since we either follow a path up the tree or a path down the tress

Analysis of Algorithms I: Binary Search Trees

Analysis of Algorithms I: Binary Search Trees Analysis of Algorithms I: Binary Search Trees Xi Chen Columbia University Hash table: A data structure that maintains a subset of keys from a universe set U = {0, 1,..., p 1} and supports all three dictionary

More information

Binary Heaps * * * * * * * / / \ / \ / \ / \ / \ * * * * * * * * * * * / / \ / \ / / \ / \ * * * * * * * * * *

Binary Heaps * * * * * * * / / \ / \ / \ / \ / \ * * * * * * * * * * * / / \ / \ / / \ / \ * * * * * * * * * * Binary Heaps A binary heap is another data structure. It implements a priority queue. Priority Queue has the following operations: isempty add (with priority) remove (highest priority) peek (at highest

More information

Binary Search Tree. 6.006 Intro to Algorithms Recitation 03 February 9, 2011

Binary Search Tree. 6.006 Intro to Algorithms Recitation 03 February 9, 2011 Binary Search Tree A binary search tree is a data structure that allows for key lookup, insertion, and deletion. It is a binary tree, meaning every node of the tree has at most two child nodes, a left

More information

Introduction to Data Structures and Algorithms

Introduction to Data Structures and Algorithms Introduction to Data Structures and Algorithms Chapter: Binary Search Trees Lehrstuhl Informatik 7 (Prof. Dr.-Ing. Reinhard German) Martensstraße 3, 91058 Erlangen Search Trees Search trees can be used

More information

Algorithms Chapter 12 Binary Search Trees

Algorithms Chapter 12 Binary Search Trees Algorithms Chapter 1 Binary Search Trees Outline Assistant Professor: Ching Chi Lin 林 清 池 助 理 教 授 chingchi.lin@gmail.com Department of Computer Science and Engineering National Taiwan Ocean University

More information

A binary search tree is a binary tree with a special property called the BST-property, which is given as follows:

A binary search tree is a binary tree with a special property called the BST-property, which is given as follows: Chapter 12: Binary Search Trees A binary search tree is a binary tree with a special property called the BST-property, which is given as follows: For all nodes x and y, if y belongs to the left subtree

More information

Balanced Binary Search Tree

Balanced Binary Search Tree AVL Trees / Slide 1 Balanced Binary Search Tree Worst case height of binary search tree: N-1 Insertion, deletion can be O(N) in the worst case We want a tree with small height Height of a binary tree with

More information

Data Structures and Algorithm Analysis (CSC317) Intro/Review of Data Structures Focus on dynamic sets

Data Structures and Algorithm Analysis (CSC317) Intro/Review of Data Structures Focus on dynamic sets Data Structures and Algorithm Analysis (CSC317) Intro/Review of Data Structures Focus on dynamic sets We ve been talking a lot about efficiency in computing and run time. But thus far mostly ignoring data

More information

CS 253: Algorithms. Chapter 12. Binary Search Trees. * Deletion and Problems. Credit: Dr. George Bebis

CS 253: Algorithms. Chapter 12. Binary Search Trees. * Deletion and Problems. Credit: Dr. George Bebis CS 2: Algorithms Chapter Binary Search Trees * Deletion and Problems Credit: Dr. George Bebis Binary Search Trees Tree representation: A linked data structure in which each node is an object Node representation:

More information

Lecture 6: Binary Search Trees CSCI 700 - Algorithms I. Andrew Rosenberg

Lecture 6: Binary Search Trees CSCI 700 - Algorithms I. Andrew Rosenberg Lecture 6: Binary Search Trees CSCI 700 - Algorithms I Andrew Rosenberg Last Time Linear Time Sorting Counting Sort Radix Sort Bucket Sort Today Binary Search Trees Data Structures Data structure is a

More information

schema binary search tree schema binary search trees data structures and algorithms 2015 09 21 lecture 7 AVL-trees material

schema binary search tree schema binary search trees data structures and algorithms 2015 09 21 lecture 7 AVL-trees material scema binary searc trees data structures and algoritms 05 0 lecture 7 VL-trees material scema binary searc tree binary tree: linked data structure wit nodes containing binary searc trees VL-trees material

More information

Binary Trees and Huffman Encoding Binary Search Trees

Binary Trees and Huffman Encoding Binary Search Trees Binary Trees and Huffman Encoding Binary Search Trees Computer Science E119 Harvard Extension School Fall 2012 David G. Sullivan, Ph.D. Motivation: Maintaining a Sorted Collection of Data A data dictionary

More information

Binary Heaps. CSE 373 Data Structures

Binary Heaps. CSE 373 Data Structures Binary Heaps CSE Data Structures Readings Chapter Section. Binary Heaps BST implementation of a Priority Queue Worst case (degenerate tree) FindMin, DeleteMin and Insert (k) are all O(n) Best case (completely

More information

Binary Search Trees. A Generic Tree. Binary Trees. Nodes in a binary search tree ( B-S-T) are of the form. P parent. Key. Satellite data L R

Binary Search Trees. A Generic Tree. Binary Trees. Nodes in a binary search tree ( B-S-T) are of the form. P parent. Key. Satellite data L R Binary Search Trees A Generic Tree Nodes in a binary search tree ( B-S-T) are of the form P parent Key A Satellite data L R B C D E F G H I J The B-S-T has a root node which is the only node whose parent

More information

Binary Search Trees. Data in each node. Larger than the data in its left child Smaller than the data in its right child

Binary Search Trees. Data in each node. Larger than the data in its left child Smaller than the data in its right child Binary Search Trees Data in each node Larger than the data in its left child Smaller than the data in its right child FIGURE 11-6 Arbitrary binary tree FIGURE 11-7 Binary search tree Data Structures Using

More information

Binary Search Trees > = Binary Search Trees 1. 2004 Goodrich, Tamassia

Binary Search Trees > = Binary Search Trees 1. 2004 Goodrich, Tamassia Binary Search Trees < 6 2 > = 1 4 8 9 Binary Search Trees 1 Binary Search Trees A binary search tree is a binary tree storing keyvalue entries at its internal nodes and satisfying the following property:

More information

Sidebar: Data Structures

Sidebar: Data Structures Sidebar: Data Structures A data structure is a collection of algorithms for storing and retrieving information. The operations that store information are called updates, and the operations that retrieve

More information

Binary Search Trees 3/20/14

Binary Search Trees 3/20/14 Binary Search Trees 3/0/4 Presentation for use ith the textbook Data Structures and Algorithms in Java, th edition, by M. T. Goodrich, R. Tamassia, and M. H. Goldasser, Wiley, 04 Binary Search Trees 4

More information

Chapter 14 The Binary Search Tree

Chapter 14 The Binary Search Tree Chapter 14 The Binary Search Tree In Chapter 5 we discussed the binary search algorithm, which depends on a sorted vector. Although the binary search, being in O(lg(n)), is very efficient, inserting a

More information

From Last Time: Remove (Delete) Operation

From Last Time: Remove (Delete) Operation CSE 32 Lecture : More on Search Trees Today s Topics: Lazy Operations Run Time Analysis of Binary Search Tree Operations Balanced Search Trees AVL Trees and Rotations Covered in Chapter of the text From

More information

1 2-3 Trees: The Basics

1 2-3 Trees: The Basics CS10: Data Structures and Object-Oriented Design (Fall 2013) November 1, 2013: 2-3 Trees: Inserting and Deleting Scribes: CS 10 Teaching Team Lecture Summary In this class, we investigated 2-3 Trees in

More information

Sorting revisited. Build the binary search tree: O(n^2) Traverse the binary tree: O(n) Total: O(n^2) + O(n) = O(n^2)

Sorting revisited. Build the binary search tree: O(n^2) Traverse the binary tree: O(n) Total: O(n^2) + O(n) = O(n^2) Sorting revisited How did we use a binary search tree to sort an array of elements? Tree Sort Algorithm Given: An array of elements to sort 1. Build a binary search tree out of the elements 2. Traverse

More information

Full and Complete Binary Trees

Full and Complete Binary Trees Full and Complete Binary Trees Binary Tree Theorems 1 Here are two important types of binary trees. Note that the definitions, while similar, are logically independent. Definition: a binary tree T is full

More information

Previous Lectures. B-Trees. External storage. Two types of memory. B-trees. Main principles

Previous Lectures. B-Trees. External storage. Two types of memory. B-trees. Main principles B-Trees Algorithms and data structures for external memory as opposed to the main memory B-Trees Previous Lectures Height balanced binary search trees: AVL trees, red-black trees. Multiway search trees:

More information

Tree Properties (size vs height) Balanced Binary Trees

Tree Properties (size vs height) Balanced Binary Trees Tree Properties (size vs height) Balanced Binary Trees Due now: WA 7 (This is the end of the grace period). Note that a grace period takes the place of a late day. Now is the last time you can turn this

More information

root node level: internal node edge leaf node CS@VT Data Structures & Algorithms 2000-2009 McQuain

root node level: internal node edge leaf node CS@VT Data Structures & Algorithms 2000-2009 McQuain inary Trees 1 A binary tree is either empty, or it consists of a node called the root together with two binary trees called the left subtree and the right subtree of the root, which are disjoint from each

More information

A binary search tree or BST is a binary tree that is either empty or in which the data element of each node has a key, and:

A binary search tree or BST is a binary tree that is either empty or in which the data element of each node has a key, and: Binary Search Trees 1 The general binary tree shown in the previous chapter is not terribly useful in practice. The chief use of binary trees is for providing rapid access to data (indexing, if you will)

More information

DNS LOOKUP SYSTEM DATA STRUCTURES AND ALGORITHMS PROJECT REPORT

DNS LOOKUP SYSTEM DATA STRUCTURES AND ALGORITHMS PROJECT REPORT DNS LOOKUP SYSTEM DATA STRUCTURES AND ALGORITHMS PROJECT REPORT By GROUP Avadhut Gurjar Mohsin Patel Shraddha Pandhe Page 1 Contents 1. Introduction... 3 2. DNS Recursive Query Mechanism:...5 2.1. Client

More information

Ordered Lists and Binary Trees

Ordered Lists and Binary Trees Data Structures and Algorithms Ordered Lists and Binary Trees Chris Brooks Department of Computer Science University of San Francisco Department of Computer Science University of San Francisco p.1/62 6-0:

More information

In our lecture about Binary Search Trees (BST) we have seen that the operations of searching for and inserting an element in a BST can be of the

In our lecture about Binary Search Trees (BST) we have seen that the operations of searching for and inserting an element in a BST can be of the In our lecture about Binary Search Trees (BST) we have seen that the operations of searching for and inserting an element in a BST can be of the order O(log n) (in the best case) and of the order O(n)

More information

Big Data and Scripting. Part 4: Memory Hierarchies

Big Data and Scripting. Part 4: Memory Hierarchies 1, Big Data and Scripting Part 4: Memory Hierarchies 2, Model and Definitions memory size: M machine words total storage (on disk) of N elements (N is very large) disk size unlimited (for our considerations)

More information

1) The postfix expression for the infix expression A+B*(C+D)/F+D*E is ABCD+*F/DE*++

1) The postfix expression for the infix expression A+B*(C+D)/F+D*E is ABCD+*F/DE*++ Answer the following 1) The postfix expression for the infix expression A+B*(C+D)/F+D*E is ABCD+*F/DE*++ 2) Which data structure is needed to convert infix notations to postfix notations? Stack 3) The

More information

B-Trees. Algorithms and data structures for external memory as opposed to the main memory B-Trees. B -trees

B-Trees. Algorithms and data structures for external memory as opposed to the main memory B-Trees. B -trees B-Trees Algorithms and data structures for external memory as opposed to the main memory B-Trees Previous Lectures Height balanced binary search trees: AVL trees, red-black trees. Multiway search trees:

More information

Binary Heap Algorithms

Binary Heap Algorithms CS Data Structures and Algorithms Lecture Slides Wednesday, April 5, 2009 Glenn G. Chappell Department of Computer Science University of Alaska Fairbanks CHAPPELLG@member.ams.org 2005 2009 Glenn G. Chappell

More information

Binary Search Trees. Ric Glassey glassey@kth.se

Binary Search Trees. Ric Glassey glassey@kth.se Binary Search Trees Ric Glassey glassey@kth.se Outline Binary Search Trees Aim: Demonstrate how a BST can maintain order and fast performance relative to its height Properties Operations Min/Max Search

More information

Lecture 2 February 12, 2003

Lecture 2 February 12, 2003 6.897: Advanced Data Structures Spring 003 Prof. Erik Demaine Lecture February, 003 Scribe: Jeff Lindy Overview In the last lecture we considered the successor problem for a bounded universe of size u.

More information

AS-2261 M.Sc.(First Semester) Examination-2013 Paper -fourth Subject-Data structure with algorithm

AS-2261 M.Sc.(First Semester) Examination-2013 Paper -fourth Subject-Data structure with algorithm AS-2261 M.Sc.(First Semester) Examination-2013 Paper -fourth Subject-Data structure with algorithm Time: Three Hours] [Maximum Marks: 60 Note Attempts all the questions. All carry equal marks Section A

More information

Data Structures and Algorithms

Data Structures and Algorithms Data Structures and Algorithms CS245-2016S-06 Binary Search Trees David Galles Department of Computer Science University of San Francisco 06-0: Ordered List ADT Operations: Insert an element in the list

More information

SOLVING DATABASE QUERIES USING ORTHOGONAL RANGE SEARCHING

SOLVING DATABASE QUERIES USING ORTHOGONAL RANGE SEARCHING SOLVING DATABASE QUERIES USING ORTHOGONAL RANGE SEARCHING CPS234: Computational Geometry Prof. Pankaj Agarwal Prepared by: Khalid Al Issa (khalid@cs.duke.edu) Fall 2005 BACKGROUND : RANGE SEARCHING, a

More information

Learning Outcomes. COMP202 Complexity of Algorithms. Binary Search Trees and Other Search Trees

Learning Outcomes. COMP202 Complexity of Algorithms. Binary Search Trees and Other Search Trees Learning Outcomes COMP202 Complexity of Algorithms Binary Search Trees and Other Search Trees [See relevant sections in chapters 2 and 3 in Goodrich and Tamassia.] At the conclusion of this set of lecture

More information

Binary Search Trees. Each child can be identied as either a left or right. parent. right. A binary tree can be implemented where each node

Binary Search Trees. Each child can be identied as either a left or right. parent. right. A binary tree can be implemented where each node Binary Search Trees \I think that I shall never see a poem as lovely as a tree Poem's are wrote by fools like me but only G-d can make atree \ {Joyce Kilmer Binary search trees provide a data structure

More information

Symbol Tables. Introduction

Symbol Tables. Introduction Symbol Tables Introduction A compiler needs to collect and use information about the names appearing in the source program. This information is entered into a data structure called a symbol table. The

More information

Binary Search Trees. basic implementations randomized BSTs deletion in BSTs

Binary Search Trees. basic implementations randomized BSTs deletion in BSTs Binary Search Trees basic implementations randomized BSTs deletion in BSTs eferences: Algorithms in Java, Chapter 12 Intro to Programming, Section 4.4 http://www.cs.princeton.edu/introalgsds/43bst 1 Elementary

More information

Binary Search Trees (BST)

Binary Search Trees (BST) Binary Search Trees (BST) 1. Hierarchical data structure with a single reference to node 2. Each node has at most two child nodes (a left and a right child) 3. Nodes are organized by the Binary Search

More information

A binary heap is a complete binary tree, where each node has a higher priority than its children. This is called heap-order property

A binary heap is a complete binary tree, where each node has a higher priority than its children. This is called heap-order property CmSc 250 Intro to Algorithms Chapter 6. Transform and Conquer Binary Heaps 1. Definition A binary heap is a complete binary tree, where each node has a higher priority than its children. This is called

More information

Questions 1 through 25 are worth 2 points each. Choose one best answer for each.

Questions 1 through 25 are worth 2 points each. Choose one best answer for each. Questions 1 through 25 are worth 2 points each. Choose one best answer for each. 1. For the singly linked list implementation of the queue, where are the enqueues and dequeues performed? c a. Enqueue in

More information

TREE BASIC TERMINOLOGIES

TREE BASIC TERMINOLOGIES TREE Trees are very flexible, versatile and powerful non-liner data structure that can be used to represent data items possessing hierarchical relationship between the grand father and his children and

More information

Classification/Decision Trees (II)

Classification/Decision Trees (II) Classification/Decision Trees (II) Department of Statistics The Pennsylvania State University Email: jiali@stat.psu.edu Right Sized Trees Let the expected misclassification rate of a tree T be R (T ).

More information

Load Balancing. Load Balancing 1 / 24

Load Balancing. Load Balancing 1 / 24 Load Balancing Backtracking, branch & bound and alpha-beta pruning: how to assign work to idle processes without much communication? Additionally for alpha-beta pruning: implementing the young-brothers-wait

More information

Output: 12 18 30 72 90 87. struct treenode{ int data; struct treenode *left, *right; } struct treenode *tree_ptr;

Output: 12 18 30 72 90 87. struct treenode{ int data; struct treenode *left, *right; } struct treenode *tree_ptr; 50 20 70 10 30 69 90 14 35 68 85 98 16 22 60 34 (c) Execute the algorithm shown below using the tree shown above. Show the exact output produced by the algorithm. Assume that the initial call is: prob3(root)

More information

Data Structures. Jaehyun Park. CS 97SI Stanford University. June 29, 2015

Data Structures. Jaehyun Park. CS 97SI Stanford University. June 29, 2015 Data Structures Jaehyun Park CS 97SI Stanford University June 29, 2015 Typical Quarter at Stanford void quarter() { while(true) { // no break :( task x = GetNextTask(tasks); process(x); // new tasks may

More information

Lecture 7: Approximation via Randomized Rounding

Lecture 7: Approximation via Randomized Rounding Lecture 7: Approximation via Randomized Rounding Often LPs return a fractional solution where the solution x, which is supposed to be in {0, } n, is in [0, ] n instead. There is a generic way of obtaining

More information

Algorithms and Data Structures

Algorithms and Data Structures Algorithms and Data Structures CMPSC 465 LECTURES 20-21 Priority Queues and Binary Heaps Adam Smith S. Raskhodnikova and A. Smith. Based on slides by C. Leiserson and E. Demaine. 1 Trees Rooted Tree: collection

More information

CS711008Z Algorithm Design and Analysis

CS711008Z Algorithm Design and Analysis CS711008Z Algorithm Design and Analysis Lecture 7 Binary heap, binomial heap, and Fibonacci heap 1 Dongbo Bu Institute of Computing Technology Chinese Academy of Sciences, Beijing, China 1 The slides were

More information

Data Structures Fibonacci Heaps, Amortized Analysis

Data Structures Fibonacci Heaps, Amortized Analysis Chapter 4 Data Structures Fibonacci Heaps, Amortized Analysis Algorithm Theory WS 2012/13 Fabian Kuhn Fibonacci Heaps Lacy merge variant of binomial heaps: Do not merge trees as long as possible Structure:

More information

Binary Coded Web Access Pattern Tree in Education Domain

Binary Coded Web Access Pattern Tree in Education Domain Binary Coded Web Access Pattern Tree in Education Domain C. Gomathi P.G. Department of Computer Science Kongu Arts and Science College Erode-638-107, Tamil Nadu, India E-mail: kc.gomathi@gmail.com M. Moorthi

More information

1/1 7/4 2/2 12/7 10/30 12/25

1/1 7/4 2/2 12/7 10/30 12/25 Binary Heaps A binary heap is dened to be a binary tree with a key in each node such that: 1. All leaves are on, at most, two adjacent levels. 2. All leaves on the lowest level occur to the left, and all

More information

CSE 326: Data Structures B-Trees and B+ Trees

CSE 326: Data Structures B-Trees and B+ Trees Announcements (4//08) CSE 26: Data Structures B-Trees and B+ Trees Brian Curless Spring 2008 Midterm on Friday Special office hour: 4:-5: Thursday in Jaech Gallery (6 th floor of CSE building) This is

More information

Outline BST Operations Worst case Average case Balancing AVL Red-black B-trees. Binary Search Trees. Lecturer: Georgy Gimel farb

Outline BST Operations Worst case Average case Balancing AVL Red-black B-trees. Binary Search Trees. Lecturer: Georgy Gimel farb Binary Search Trees Lecturer: Georgy Gimel farb COMPSCI 220 Algorithms and Data Structures 1 / 27 1 Properties of Binary Search Trees 2 Basic BST operations The worst-case time complexity of BST operations

More information

6 March 2007 1. Array Implementation of Binary Trees

6 March 2007 1. Array Implementation of Binary Trees Heaps CSE 0 Winter 00 March 00 1 Array Implementation of Binary Trees Each node v is stored at index i defined as follows: If v is the root, i = 1 The left child of v is in position i The right child of

More information

Introduction Advantages and Disadvantages Algorithm TIME COMPLEXITY. Splay Tree. Cheruku Ravi Teja. November 14, 2011

Introduction Advantages and Disadvantages Algorithm TIME COMPLEXITY. Splay Tree. Cheruku Ravi Teja. November 14, 2011 November 14, 2011 1 Real Time Applications 2 3 Results of 4 Real Time Applications Splay trees are self branching binary search tree which has the property of reaccessing the elements quickly that which

More information

EE602 Algorithms GEOMETRIC INTERSECTION CHAPTER 27

EE602 Algorithms GEOMETRIC INTERSECTION CHAPTER 27 EE602 Algorithms GEOMETRIC INTERSECTION CHAPTER 27 The Problem Given a set of N objects, do any two intersect? Objects could be lines, rectangles, circles, polygons, or other geometric objects Simple to

More information

Persistent Data Structures

Persistent Data Structures 6.854 Advanced Algorithms Lecture 2: September 9, 2005 Scribes: Sommer Gentry, Eddie Kohler Lecturer: David Karger Persistent Data Structures 2.1 Introduction and motivation So far, we ve seen only ephemeral

More information

Converting a Number from Decimal to Binary

Converting a Number from Decimal to Binary Converting a Number from Decimal to Binary Convert nonnegative integer in decimal format (base 10) into equivalent binary number (base 2) Rightmost bit of x Remainder of x after division by two Recursive

More information

Algorithms and Data Structures

Algorithms and Data Structures Algorithms and Data Structures Part 2: Data Structures PD Dr. rer. nat. habil. Ralf-Peter Mundani Computation in Engineering (CiE) Summer Term 2016 Overview general linked lists stacks queues trees 2 2

More information

Heap. Binary Search Tree. Heaps VS BSTs. < el el. Difference between a heap and a BST:

Heap. Binary Search Tree. Heaps VS BSTs. < el el. Difference between a heap and a BST: Heaps VS BSTs Difference between a heap and a BST: Heap el Binary Search Tree el el el < el el Perfectly balanced at all times Immediate access to maximal element Easy to code Does not provide efficient

More information

Exam study sheet for CS2711. List of topics

Exam study sheet for CS2711. List of topics Exam study sheet for CS2711 Here is the list of topics you need to know for the final exam. For each data structure listed below, make sure you can do the following: 1. Give an example of this data structure

More information

UNIVERSITI MALAYSIA SARAWAK KOTA SAMARAHAN SARAWAK PSD2023 ALGORITHM & DATA STRUCTURE

UNIVERSITI MALAYSIA SARAWAK KOTA SAMARAHAN SARAWAK PSD2023 ALGORITHM & DATA STRUCTURE STUDENT IDENTIFICATION NO UNIVERSITI MALAYSIA SARAWAK 94300 KOTA SAMARAHAN SARAWAK FAKULTI SAINS KOMPUTER & TEKNOLOGI MAKLUMAT (Faculty of Computer Science & Information Technology) Diploma in Multimedia

More information

A Comparison of Dictionary Implementations

A Comparison of Dictionary Implementations A Comparison of Dictionary Implementations Mark P Neyer April 10, 2009 1 Introduction A common problem in computer science is the representation of a mapping between two sets. A mapping f : A B is a function

More information

Cpt S 223. School of EECS, WSU

Cpt S 223. School of EECS, WSU Priority Queues (Heaps) 1 Motivation Queues are a standard mechanism for ordering tasks on a first-come, first-served basis However, some tasks may be more important or timely than others (higher priority)

More information

Binary Search Trees CMPSC 122

Binary Search Trees CMPSC 122 Binary Search Trees CMPSC 122 Note: This notes packet has significant overlap with the first set of trees notes I do in CMPSC 360, but goes into much greater depth on turning BSTs into pseudocode than

More information

OPTIMAL BINARY SEARCH TREES

OPTIMAL BINARY SEARCH TREES OPTIMAL BINARY SEARCH TREES 1. PREPARATION BEFORE LAB DATA STRUCTURES An optimal binary search tree is a binary search tree for which the nodes are arranged on levels such that the tree cost is minimum.

More information

Scheduling. Open shop, job shop, flow shop scheduling. Related problems. Open shop, job shop, flow shop scheduling

Scheduling. Open shop, job shop, flow shop scheduling. Related problems. Open shop, job shop, flow shop scheduling Scheduling Basic scheduling problems: open shop, job shop, flow job The disjunctive graph representation Algorithms for solving the job shop problem Computational complexity of the job shop problem Open

More information

Lecture 1: Course overview, circuits, and formulas

Lecture 1: Course overview, circuits, and formulas Lecture 1: Course overview, circuits, and formulas Topics in Complexity Theory and Pseudorandomness (Spring 2013) Rutgers University Swastik Kopparty Scribes: John Kim, Ben Lund 1 Course Information Swastik

More information

A binary search tree or BST is a binary tree that is either empty or in which the data element of each node has a key, and:

A binary search tree or BST is a binary tree that is either empty or in which the data element of each node has a key, and: 1 The general binary tree shown in the previous chapter is not terribly useful in practice. The chief use of binary trees is for providing rapid access to data (indexing, if you will) and the general binary

More information

Home Page. Data Structures. Title Page. Page 1 of 24. Go Back. Full Screen. Close. Quit

Home Page. Data Structures. Title Page. Page 1 of 24. Go Back. Full Screen. Close. Quit Data Structures Page 1 of 24 A.1. Arrays (Vectors) n-element vector start address + ielementsize 0 +1 +2 +3 +4... +n-1 start address continuous memory block static, if size is known at compile time dynamic,

More information

12 Binary Search Trees

12 Binary Search Trees 12 Binary Search Trees The search tree data structure supports many dynamic-set operations, including SEARCH, MINIMUM, MAXIMUM, PREDECESSOR, SUCCESSOR, INSERT, and DELETE. Thus, we can use a search tree

More information

External Memory Geometric Data Structures

External Memory Geometric Data Structures External Memory Geometric Data Structures Lars Arge Department of Computer Science University of Aarhus and Duke University Augues 24, 2005 1 Introduction Many modern applications store and process datasets

More information

Persistent Data Structures and Planar Point Location

Persistent Data Structures and Planar Point Location Persistent Data Structures and Planar Point Location Inge Li Gørtz Persistent Data Structures Ephemeral Partial persistence Full persistence Confluent persistence V1 V1 V1 V1 V2 q ue V2 V2 V5 V2 V4 V4

More information

Why Use Binary Trees?

Why Use Binary Trees? Binary Search Trees Why Use Binary Trees? Searches are an important application. What other searches have we considered? brute force search (with array or linked list) O(N) binarysearch with a pre-sorted

More information

Optimizations. Optimization Safety. Optimization Safety. Control Flow Graphs. Code transformations to improve program

Optimizations. Optimization Safety. Optimization Safety. Control Flow Graphs. Code transformations to improve program Optimizations Code transformations to improve program Mainly: improve execution time Also: reduce program size Control low Graphs Can be done at high level or low level E.g., constant folding Optimizations

More information

Exercises Software Development I. 11 Recursion, Binary (Search) Trees. Towers of Hanoi // Tree Traversal. January 16, 2013

Exercises Software Development I. 11 Recursion, Binary (Search) Trees. Towers of Hanoi // Tree Traversal. January 16, 2013 Exercises Software Development I 11 Recursion, Binary (Search) Trees Towers of Hanoi // Tree Traversal January 16, 2013 Software Development I Winter term 2012/2013 Institute for Pervasive Computing Johannes

More information

Optimal Binary Search Trees Meet Object Oriented Programming

Optimal Binary Search Trees Meet Object Oriented Programming Optimal Binary Search Trees Meet Object Oriented Programming Stuart Hansen and Lester I. McCann Computer Science Department University of Wisconsin Parkside Kenosha, WI 53141 {hansen,mccann}@cs.uwp.edu

More information

CS 2112 Spring 2014. 0 Instructions. Assignment 3 Data Structures and Web Filtering. 0.1 Grading. 0.2 Partners. 0.3 Restrictions

CS 2112 Spring 2014. 0 Instructions. Assignment 3 Data Structures and Web Filtering. 0.1 Grading. 0.2 Partners. 0.3 Restrictions CS 2112 Spring 2014 Assignment 3 Data Structures and Web Filtering Due: March 4, 2014 11:59 PM Implementing spam blacklists and web filters requires matching candidate domain names and URLs very rapidly

More information

The ADT Binary Search Tree

The ADT Binary Search Tree The ADT Binary Search Tree The Binary Search Tree is a particular type of binary tree that enables easy searching for specific items. Definition The ADT Binary Search Tree is a binary tree which has an

More information

International Journal of Software and Web Sciences (IJSWS) www.iasir.net

International Journal of Software and Web Sciences (IJSWS) www.iasir.net International Association of Scientific Innovation and Research (IASIR) (An Association Unifying the Sciences, Engineering, and Applied Research) ISSN (Print): 2279-0063 ISSN (Online): 2279-0071 International

More information

Introduction to Data Structures and Algorithms

Introduction to Data Structures and Algorithms Introduction to Data Structures and Algorithms Chapter: Elementary Data Structures(1) Lehrstuhl Informatik 7 (Prof. Dr.-Ing. Reinhard German) Martensstraße 3, 91058 Erlangen Overview on simple data structures

More information

Any two nodes which are connected by an edge in a graph are called adjacent node.

Any two nodes which are connected by an edge in a graph are called adjacent node. . iscuss following. Graph graph G consist of a non empty set V called the set of nodes (points, vertices) of the graph, a set which is the set of edges and a mapping from the set of edges to a set of pairs

More information

R-trees. R-Trees: A Dynamic Index Structure For Spatial Searching. R-Tree. Invariants

R-trees. R-Trees: A Dynamic Index Structure For Spatial Searching. R-Tree. Invariants R-Trees: A Dynamic Index Structure For Spatial Searching A. Guttman R-trees Generalization of B+-trees to higher dimensions Disk-based index structure Occupancy guarantee Multiple search paths Insertions

More information

A First Investigation of Sturmian Trees

A First Investigation of Sturmian Trees A First Investigation of Sturmian Trees Jean Berstel 2, Luc Boasson 1 Olivier Carton 1, Isabelle Fagnot 2 1 LIAFA, CNRS Université Paris 7 2 IGM, CNRS Université de Marne-la-Vallée Atelier de Combinatoire,

More information

Binary Search. Search for x in a sorted array A.

Binary Search. Search for x in a sorted array A. Divide and Conquer A general paradigm for algorithm design; inspired by emperors and colonizers. Three-step process: 1. Divide the problem into smaller problems. 2. Conquer by solving these problems. 3.

More information

Quiz 1 Solutions. (a) T F The height of any binary search tree with n nodes is O(log n). Explain:

Quiz 1 Solutions. (a) T F The height of any binary search tree with n nodes is O(log n). Explain: Introduction to Algorithms March 9, 2011 Massachusetts Institute of Technology 6.006 Spring 2011 Professors Erik Demaine, Piotr Indyk, and Manolis Kellis Quiz 1 Solutions Problem 1. Quiz 1 Solutions True

More information

Efficient Interval Management in Microsoft SQL Server

Efficient Interval Management in Microsoft SQL Server Efficient Interval Management in Microsoft SQL Server Itzik Ben-Gan, SolidQ Last modified: August, 0 Agenda Inefficiencies of classic interval handling Solution: RI-tree by Kriegel, Pötke and Seidl of

More information

Lecture 6: Trees, Binary Trees and Binary Search Trees (BST)

Lecture 6: Trees, Binary Trees and Binary Search Trees (BST) Lecture 6: Trees, Binary Trees and Binary Search Trees (BST) Reading materials Dale, Joyce, Weems: Chapter 8 OpenDSA: Chapter 9 Liang: only in Comprehensive edition, Binary search trees visualizations:

More information

Algorithms and Data Structures Written Exam Proposed SOLUTION

Algorithms and Data Structures Written Exam Proposed SOLUTION Algorithms and Data Structures Written Exam Proposed SOLUTION 2005-01-07 from 09:00 to 13:00 Allowed tools: A standard calculator. Grading criteria: You can get at most 30 points. For an E, 15 points are

More information

Motivation Suppose we have a database of people We want to gure out who is related to whom Initially, we only have a list of people, and information a

Motivation Suppose we have a database of people We want to gure out who is related to whom Initially, we only have a list of people, and information a CSE 220: Handout 29 Disjoint Sets 1 Motivation Suppose we have a database of people We want to gure out who is related to whom Initially, we only have a list of people, and information about relations

More information

Dynamic 3-sided Planar Range Queries with Expected Doubly Logarithmic Time

Dynamic 3-sided Planar Range Queries with Expected Doubly Logarithmic Time Dynamic 3-sided Planar Range Queries with Expected Doubly Logarithmic Time Gerth Stølting Brodal 1, Alexis C. Kaporis 2, Spyros Sioutas 3, Konstantinos Tsakalidis 1, Kostas Tsichlas 4 1 MADALGO, Department

More information

External Sorting. Chapter 13. Database Management Systems 3ed, R. Ramakrishnan and J. Gehrke 1

External Sorting. Chapter 13. Database Management Systems 3ed, R. Ramakrishnan and J. Gehrke 1 External Sorting Chapter 13 Database Management Systems 3ed, R. Ramakrishnan and J. Gehrke 1 Why Sort? A classic problem in computer science! Data requested in sorted order e.g., find students in increasing

More information

Rotation Operation for Binary Search Trees Idea:

Rotation Operation for Binary Search Trees Idea: Rotation Operation for Binary Search Trees Idea: Change a few pointers at a particular place in the tree so that one subtree becomes less deep in exchange for another one becoming deeper. A sequence of

More information

Longest common extensions in trees

Longest common extensions in trees Longest common extensions in trees Philip Bille 1 Paweł Gawrychowski 2 Inge Li Gørtz 1 Gad M. Landau 3,4 Oren Weimann 3 1 DTU Compute 2 University of Warsaw (supported by WCMCS) 3 University of Haifa 4

More information